Order of Operations

Parentheses

In the order of operations, calculations inside parentheses or brackets are performed first. This is because parentheses and brackets are used to group calculations that should be done before any other functions in the expression. 

For example, consider the following expression:

Without any parentheses, it is not clear what should be calculated first. Should the multiplication be done before the addition or the other way around? To specify the order in which the calculations should be performed, we can use parentheses:

In this case, the expression inside the parentheses, 4 + 3, is calculated first, giving a result of 7. Then the multiplication is performed, showing a final result of 35 – 2, or 33.

Using parentheses correctly in an expression is essential to ensure that the intended order of operations is followed and the correct result is obtained.

Another example to show that groupings inside should be solved first.

Step 1: In this case, the expression inside the parentheses (32 ÷ 4) should be calculated first. So it will be 8.

Step 2: Now the expression inside the square parentheses [8 + 3] should be solved. So, it will be 11.

Step 3: Then the multiplication is performed, showing a final result of 11 × 2 = 22.

Exponents

In the order of operations, calculations involving exponents, also known as powers or indices, are performed before any multiplications or divisions. This is because exponents indicate the number of times a number should be multiplied by itself, and these calculations should be done before any other operations (except parentheses)  in the expression.

An exponent is a small number written above and to the right of another number, called the base. The exponent tells you how many times to multiply the base by itself. For example, the base 4 and exponent 3 can be written like this: 4^3. This means to multiply 4 by itself 3 times, or 4*4*4. The result is 64.

For example, consider the following expression:

2 + 3 * 4^2

In this case, the exponent, 4^2, indicates that 4 should be multiplied by itself 2 times, or 4*4, which gives a result of 16. The exponent calculation is done before the multiplication, so the outcome of the expression is 2 + 3 * 16, or 2 + 48, or 50.

Multiplication and Division

In the order of operations, multiplication and division are performed before addition and subtraction. This is because multiplications and divisions are considered equal priorities and should be performed before any additions or subtractions.

For example, consider the following expression:

2 + 4 * 3 – 6 / 2

In this case, the multiplication and division should be performed before the additions and subtractions. First, the multiplication is done: 4 * 3 = 12. Then the division is done: 6 / 2 = 3. Finally, the additions and subtractions are made: 2 + 12 – 3 = 11.

Here is another example:

5 – 4 / 2 + 8 * 3

In this case, the division and multiplication should be done first: 4 / 2 = 2 and 8 * 3 = 24. Then the additions and subtractions are done: 5 – 2 + 24 = 27.

Addition and Subtraction

In the order of operations, additions and subtractions are the final calculations that should be performed. This is because they are considered equal priority and should be performed after all other calculations have been completed.

For example, consider the following expression:

2 + 4 * 3 – 6 / 2

In this case, multiplication and division should be performed before the additions and subtractions. First, the expansion is done: 4 * 3 = 12. Then the division is done: 6 / 2 = 3. Finally, the additions and subtractions are made: 2 + 12 – 3 = 11.

Here is another example:

5 – 4 / 2 + 8 * 3 

In this case, the division and multiplication should be done first: 4 / 2 = 2 and 8 * 3 = 24. Then the additions and subtractions are done: 5 – 2 + 24 = 27.

Example 1:

Solve: 2 + 6 × (4 + 5) ÷ 3 – 5 using PEMDAS.

Solution:

Step 1 – Parentheses : 2+6 × (4 + 5) ÷ 3 – 5 = 2 + 6 × 9 ÷ 3 – 5

Step 2 – Multiplication: 2 + 6 × 9 ÷ 3 – 5 = 2 + 54 ÷ 3 – 5

Step 3 – Division: 2 + 54 ÷ 3 – 5 = 2 + 18 – 5

Step 4 – Addition: 2 + 18 – 5 = 20 – 5

Step 5 – Subtraction: 20 – 5 = 15

Practice Questions:

1. Solve 100 ÷ (6 + 7 × 2) – 5 using PEMDAS.
2. Solve 600 ÷ (44 + 14 × 4) – 5 using PEMDAS.